A very simple model of the effect of breaking surface waves in the ocean, or grid stirring in laboratory experiments, is that of random forcing at the surface of a deep homogeneous fluid with linear, frictionless dynamics. The 'turbulence' is physically separated from the gravity waves by a 'rigid lid' approximation. The model predicts that the turbulent kinetic energy decreases as the inverse square of the depth beyond a depth Δ, which is the width of a typical burst (or a few grid intervals in the laboratory). The integral length scale is proportional to depth (again past Δ), so the turbulent Reynolds number is independent of depth. The model nontrivially satisfies an equation previously interpreted as giving the flux of turbulent energy in grid-stirring experiments. The measure of 'inertial dissipation' used there is thus called into question.